Physics with an added quantum of uncertainty

It seems that this is the year that black hole physics is making a splash—in addition to dark matter, black hole talks seemed to be everywhere at the FOM conference. Appropriately enough, I was sucked right in to these talks. It seems that since Erik Verlinde confused us all five years ago, a lot of progress has been made. In particular, it feels as if the presenters are far more confident about what they can do with the tricks they’ve been developing.

One sign of the progress is that the session titled “The quantum information nature of spacetime” gave me a feeling other than overwhelming confusion. The entire session was focused on the quantum nature of black holes and how the conflict between general relativity and quantum mechanics was highlighted by black holes. This is not because of the singularity at the center of the black hole but because of what happens at the event horizon.

Is that some quantum in your bent space

Before we get to that topic, I’ll quickly outline the tool that physicists use to examine this idea. It turns out that there is a deep correspondence between gravity and quantum theory. At least, under the right circumstances there is a correspondence. In fact, you need a set of rather unusual circumstances for this idea to work. There has to be a negative cosmological constant, which means that the expansion of the Universe is slowing rather than accelerating. Quantum mechanics also has to operate in one dimension more than gravity.

Under these conditions, one can map the dynamics of gravity to quantum properties in some sense—if you want to know more, you’ll have to ask someone who actually knows what they are talking about. Despite my complete lack of understanding, there are still some mind-blowing conclusions that fall out.

Rob Myers (from the Perimeter Institute) and Kyriakos Papadodimas (from the University of Groningen) showed how they could use this correspondence to understand the entropy of a black hole in a more sensible fashion. Entropy is one of the more slippery concepts in physics. Let me give a simple example: consider two magnets sitting in a magnetic field. The magnets have a total of four possible arrangements. Both magnets can arrange themselves “against” the field (so they point such that their north pole is pointing toward the applied magnetic field’s south pole). Alternatively, they can both arrange themselves to be pointing with the field (so the north poles are all in the same direction). Or the two magnets can be pointing in opposite directions—there are two possible ways to achieve this outcome.

You can also calculate that the first state has the lowest total energy and the second the highest total energy, while the last two have the same intermediate amount of energy. The entropy is a count of the number of possible arrangements that lead to the same total energy. In this example, the first and second arrangements have lower entropy than the last two, since there’s only one possible arrangement for them.

In a black hole, this straightforward counting procedure also gives you an answer. But the answer only makes sense if the black hole also has a temperature associated with it. However, a temperature requires radiation, and black holes were not supposed to radiate. That is, until Hawking came along.

The quantum is in your hole

Particles called Hawking radiation are emitted from the event horizon of a black hole—the radius at which the gravitational pull is strong enough to prevent light from escaping.

Professor Stephen Hawking

Hawking radiation is generated by a quantum process (there is no classical way to get radiation from a black hole): two virtual particles are created inside the event horizon. One escapes the event horizon via quantum tunneling, and it has positive energy and mass; the other falls into the black hole with negative energy.

As a result of this negative energy, the black hole loses mass while other particles escape as Hawking radiation. But those two particles are linked: their masses, energies, and flight direction are correlated, as are the momentum and angular momentum. Essentially, in the space where the two particles were created, there was nothing. So the sum total of all the properties of these two particles has to be nothing. If one has positive angular momentum, the other must have negative.

Except it’s not quite so simple: the creation process doesn’t specify that particle one will have, for instance, positive angular momentum and particle two will have negative angular momentum. Instead, the creation process says that each particle has both positive and negative angular momenta at the same time. In the language of quantum mechanics, each photon of Hawking radiation is in a quantum superposition of two angular momentum states, and its angular momentum state is entangled with its partner inside the black hole.

That is, if we measure the angular momentum of the Hawking radiation, we set its state to some value. In doing so, we also set the state of the particle inside the black hole. This entanglement is a problem because it implies that the inside of a black hole is highly correlated with the outside universe.

This idea is highlighted by measuring the entropy of a black hole via correlations. Let’s go back to our magnets and consider a slightly larger group of, say, ten magnets. Another way to measure the entropy would be to divide the magnets into two groups of five magnets and measure how well correlated the two groups are. If we find that whenever the third magnet in group one flipped direction, the fourth magnet in the second group also flipped, that sequence automatically limits the total number of possible arrangements.

Put these concepts together and we could conclude that the inside of a black hole is highly correlated. Consequently, Hawking radiation is not thermal radiation, like light from the sun. Each photon should be entangled with all the other photons emitted (including those emitted in the past). At the same time, each photon is entangled with its partner photon, which is still falling into the black hole.

This situation is difficult to swallow. In quantum mechanics, you cannot independently entangle a particle to two other particles. The basic rule is that you can’t entangle particle A with particle B and then choose to entangle particle A with particle C without breaking the entanglement between A and B. You can simultaneously entangle all three particles, but that’s not what is going on with Hawking radiation.

The upshot is that if we want to get a sensible entropy for a black hole, then we require the correlations inside a black hole to have a certain flavor. But that flavor changes the nature of Hawking radiation, violating some deeply held physical principles.

Commutators and consequences

The main conclusion from Myers is that entanglement is actually a consequence of the nature of spacetime. I’m pretty sure I don’t know what that means, but it’s still exciting. Currently, in quantum mechanics, space and time are a backdrop on which everything happens. In general relativity, they are inextricably caught up in the action. We see hints that the same is true in quantum mechanics—that spacetime is not a passive stage but imprints itself into the most fundamental of quantum phenomena.